Far from equilibrium steady states of 1D-Schrödinger-Poisson systems with quantum wells I
Annales de l'I.H.P. Analyse non linéaire, Volume 25 (2008) no. 5, p. 937-968
@article{AIHPC_2008__25_5_937_0,
     author = {Bonnaillie-No\"el, V. and Nier, F. and Patel, Y.},
     title = {Far from equilibrium steady states of 1D-Schr\"odinger-Poisson systems with quantum wells I},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {25},
     number = {5},
     year = {2008},
     pages = {937-968},
     doi = {10.1016/j.anihpc.2007.05.007},
     zbl = {1149.82349},
     mrnumber = {2457818},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2008__25_5_937_0}
}
Bonnaillie-Noël, V.; Nier, F.; Patel, Y. Far from equilibrium steady states of 1D-Schrödinger-Poisson systems with quantum wells I. Annales de l'I.H.P. Analyse non linéaire, Volume 25 (2008) no. 5, pp. 937-968. doi : 10.1016/j.anihpc.2007.05.007. http://www.numdam.org/item/AIHPC_2008__25_5_937_0/

[1] Balslev E., Combes J.M., Spectral properties of many-body Schrödinger operators with dilatation-analytic interactions, Commun Math. Phys. 22 (1971) 280-294. | MR 345552 | Zbl 0219.47005

[2] Baro M., Kaiser H.-Chr., Neidhardt H., Rehberg J., A quantum transmitting Schrödinger-Poisson system, Rev. Math. Phys. 16 (3) (2004) 281-330. | MR 2063049 | Zbl 1134.82336

[3] Baro M., Kaiser H.-Chr., Neidhardt H., Rehberg J., Dissipative Schrödinger-Poisson systems, J. Math. Phys. 45 (1) (2004) 21-43. | MR 2026356 | Zbl 1070.82019

[4] Ben Abdallah N., Degond P., Markowich P.A., On a one-dimensional Schrödinger-Poisson scattering model, Z. Angew. Math. Phys. 48 (1) (1997) 135-155. | MR 1439739 | Zbl 0885.34067

[5] Bonnaillie-Noël V., Nier F., Patel M., Computing the steady states for an asymptotic model of quantum transport in resonant heterostructures, J. Comp. Phys. 219 (2006) 644-670. | MR 2274952 | Zbl pre05080109

[6] V. Bonnaillie-Noël, F. Nier, M. Patel, Far from equilibrium steady states of 1D-Schrödinger-Poisson systems with quantum wells II, Prépublications IRMAR, 2007. | Zbl 1157.82046

[7] Büttiker M., Imry Y., Landauer R., Pinhas S., Generalized many-channel conductance formula with application to small rings, Phys. Rev. B 31 (1985) 6207-6215.

[8] Chevoir F., Vinter B., Scattering assisted tunneling in double barriers diode: scattering rates and valley current, Phys. Rev. B 47 (1993) 7260-7274.

[9] Davies E.B., Spectral Theory and Differential Operators, Cambridge Studies in Advanced Mathematics, vol. 42, Cambridge University Press, Cambridge, 1995. | MR 1349825 | Zbl 0893.47004

[10] Degond P., Mehats F., Ringhofer C., Quantum hydrodynamic models derived from the entropy principle, Contemp. Math. 371 (2005) 107-131. | MR 2143862 | Zbl 1080.35152

[11] Derezinski J., Gérard C., Asymptotic Completeness of Classical and Quantum N-Particles Systems, Texts and Monographs in Physics, Springer-Verlag, 1997.

[12] Dimassi M., Sjöstrand J., Spectral Asymptotics in the Semi-Classical Limit, London Mathematical Society Lecture Note Series, vol. 268, Cambridge University Press, 1999. | MR 1735654 | Zbl 0926.35002

[13] Gérard C., Martinez A., Semiclassical asymptotics for the spectral function of long-range Schrödinger operators, J. Funct. Anal. 84 (1) (1989) 226-254. | MR 999499 | Zbl 0692.35069

[14] Helffer B., Semi-Classical Analysis for the Schrödinger Operator and Applications, Lecture Notes in Mathematics, vol. 1336, Springer-Verlag, 1988. | MR 960278 | Zbl 0647.35002

[15] Helffer B., Sjöstrand J., Résonances en limite semi-classique, Mém. Soc. Math. France 24-25 (1986). | Numdam | Zbl 0631.35075

[16] Helffer B., Sjöstrand J., Multiple wells in the semi-classical limit I, Comm. Partial Differential Equations 9 (4) (1984) 337-408. | MR 740094 | Zbl 0546.35053

[17] Helffer B., Sjöstrand J., Puits Multiples en limite semi-classique II. Interaction moléculaire. Symétries. Perturbation, Ann. Inst. H. Poincaré Phys. Théor. 42 (2) (1985) 127-212. | Numdam | MR 798695 | Zbl 0595.35031

[18] Helffer B., Sjöstrand J., Analyse semiclassique pour l'équation de Harper, Mém. Soc. Math. France 34 (1988). | Numdam | Zbl 0714.34130

[19] Hislop P.D., Sigal I.M., Introduction to Spectral Theory with Applications to Schrödinger Operators, Applied Mathematical Sciences, vol. 113, Springer-Verlag, New York, 1996. | MR 1361167 | Zbl 0855.47002

[20] Jakšić V., Pillet C.-A., Non-equilibrium steady states of finite quantum systems coupled to thermal reservoirs, Commun. Math. Phys. 226 (1) (2002) 131-162. | MR 1889995 | Zbl 0990.82017

[21] Jona-Lasinio G., Presilla C., Sjöstrand J., On Schrödinger equations with concentrated nonlinearities, Ann. Phys. 240 (1) (1995) 1-21. | MR 1329589 | Zbl 0820.34050

[22] Kastrup J., Klann R., Grahn H., Ploog K., Bonilla L., Galàn J., Kindelan M., Moscoso M., Merlin R., Self-oscillations of domains in doped GaAs-Al-As superlatices, Phys. Rev. B 52 19 (1995) 13761-13764.

[23] Landauer R., Spatial variation of currents and fields due to localized scatterers in metallic conduction, IBM J. Res. Develop. 1 (1957) 223-231. | MR 90369

[24] Nier F., A variational formulation of Schrödinger-Poisson systems in dimension d3, Comm. Partial Differential Equations 18 (7-8) (1993) 1125-1147. | MR 1233187 | Zbl 0785.35086

[25] Nier F., Schrödinger-Poisson systems in dimension d3: the whole-space case, Proc. Roy. Soc. Edinburgh Sect. A 123 (6) (1993) 1179-1201. | MR 1263914 | Zbl 0807.35119

[26] Nier F., The dynamics of some quantum open systems with short-range nonlinearities, Nonlinearity 11 (4) (1998) 1127-1172. | MR 1632618 | Zbl 0909.34052

[27] Nier F., Patel M., Nonlinear asymptotics for quantum out-of-equilibrium 1D systems: reduced models and algorithms, in: Blanchard , Dell'Antonio (Eds.), Multiscale Methods in Quantum Mechanics: Theory and Experiment, Birkhäuser, 2004, pp. 99-111. | MR 2089718

[28] M. Patel, Développement de modèles macroscopiques pour des systèmes quantiques non-linéaires hors-équilibre, Ph.D. Thesis, Université de Rennes 1, 2005.

[29] Presilla C., Sjöstrand J., Transport properties in resonant tunneling heterostructures, J. Math. Phys. 37 (10) (1996) 4816-4844. | MR 1411610 | Zbl 0868.35112

[30] Simon B., Trace Ideals and Their Applications, London Mathematical Society Lecture Note Series, vol. 35, Cambridge University Press, 1979. | MR 541149 | Zbl 0423.47001

[31] J. Sjöstrand, M. Zworski, Elementary linear algebra for advanced spectral problems, http://math.berkeley.edu/~zworsky/.

[32] Yafaev D., Mathematical Scattering Theory, General Theory, Translation of Mathematical Monographs, vol. 105, Amer. Math. Soc., 1992. | MR 1180965 | Zbl 0761.47001